In the electrical or optical transmission of digital information by digital modulation, the transmitter typically encodes the data bits of the digital information and filters sequences of the data bits to form pulses for transmission through a channel, and the receiver typically takes digital samples of the received pulses and filters the digital samples to retrieve or regenerate the original digital information. As unambiguously asserted by theory, the information-bearing waveforms used for transmission of information in communication systems ought to satisfy the so called Nyquist criterion so as to avail communication without intersymbol interference (ISI). In practice, the family of waveforms used almost exclusively are the raised-cosine and root-raised cosine pulses, often interchangeably, although strictly speaking erroneously denoted as the Nyquist pulses. The ideal Nyquist pulses would be infinite in length. However, in practical implementations, i.e., in real systems, the pulses have to be truncated at the transmitter and sampled at a finite number of points or intervals at the receiver. To do this, the filter functions in the transmitter and receiver use filters referred to as finite impulse response (FIR) filters. The finite aspect of the filters both affects the spectral shape of the transmitted pulses and the effectiveness of the subsequent filtering response by the receiver. The result of the response truncation is departure from ideal Nyquist pulses and generally appearance of the ISI, which refers to distortions of a transmission signal in which one symbol (represented by a pulse) interferes with other adjacent or nearby symbols. In addition to the intended pulse shaping, a signal usually picks up various impairments, distortions and noise when passing through the transmission channel.
FIGS. 1 and 2 illustrate the general effect of ISI. FIG. 1 shows an example graph 101 of transmission pulses (for two bits 1,0) produced by a transmitter and a graph 102 of reception pulses received by a receiver. The transmission pulses in this example are idealized as square waves, but in reality the vertical edges (i.e., edges of the signal band) have a finite slope. The reception pulses, on the other hand, tend to get elongated and smeared out. As long as the reception pulses do not overlap, there is no ISI. Thus, the receiver can sample the reception signal at any point within the same pulse intervals of the original transmission pulses (as indicated by dashed lines) and produce the correct data. However, FIG. 2 shows an example graph 201 of transmission pulses (for five bits 1,0,1,1,0) produced by a transmitter, a graph 202 of the reception pulses for each transmission pulse, and a graph 203 of a reception signal received by a receiver. In this case, the elongated and smeared out reception pulses overlap, so that the net effect detected by the receiver is the irregular reception signal of graph 203. Thus, as long as the receiver samples the reception signal at proper locations, e.g., as indicated by the dots, then the correct data will be obtained. However, if the receiver samples the reception signal within the subintervals indicated by arrows 204-206, the receiver will obtain incorrect data, even though the sample locations would be within the correct pulse intervals of the original transmission pulses. The potential for generating incorrect data is the overall issue that must be avoided or minimized.
ISI can be caused by many different reasons. For example, it can be caused by filtering effects from hardware or frequency selective fading, multipath interference, from non-linearities, and from charging/discharging effects. Very few systems are immune to ISI, so it is nearly always present in communication systems. Thus, communication system designs nearly always need to incorporate some way of controlling, mitigating or minimizing ISI.
One of the simplest solutions for reducing ISI is to simply slow down the transmission rate of the signal that is passed through the channel, e.g., with a delay between each pulse as illustrated by FIG. 1. Thus, the next pulse of information is transmitted only after allowing the current received pulse to damp down, so that the subsequent pulse does not interfere with the current pulse. Slowing down the transmission rate, however, is an easy, but unacceptable, solution. Instead, it is desired to be able to transmit the pulses at a much higher rate, as illustrated by FIG. 2.
To provide the best transmission rate through the channel, ISI generally has to be minimized without providing a delay between transmission pulses. To be able to handle the higher transmission rate, the primary techniques used to counter ISI involve “pulse-shaping.” Pulse-shaping techniques generally modulate the pulses with a particular shape at the transmitter, and use digital demodulation processes at the receiver, in such a manner that the points at which the pulses are sampled are only minimally affected by interference.
The square (or almost square) pulse shapes in the examples of FIGS. 1 and 2 are generally inadequate for pulse shaping purposes, as they cannot be accomplished in practice. Instead, the pulse shaping is commonly based on other forms, such as a sinc pulse, a raised-cosine (RC) (or root-raised-cosine (RRC)) pulse, or a Nyquist pulse, as illustrated by an ideal RC/RRC time domain pulse 301 in FIG. 3. The oscillations in the left and right side tails of the time domain pulse 301 are slowly diminished, but never truly die out, which is illustrative of the infinite length of the ideal pulse mentioned above. The dots, on the other hand, represent sampling locations (or tap points) used for finite generation (synthesis) and analysis of the pulses. Any number of taps can be used, e.g., 16, 32, 48, 64, 128. Generally, a larger number of more tightly spaced tap points provides a higher quality result. However, the larger number of tap points also generally requires more complicated, or costly hardware and/or higher power dissipation of the associated hardware.
To minimize ISI, therefore, the transmitter and receiver commonly use raised-cosine filters or root-raised-cosine filters to shape the pulses at the transmitter and handle the response at the receiver. Variations on each type of filter, however, result in pulses with different shapes. In order to get the best results for minimizing ISI, therefore, it is generally accepted that the filter in the receiver must match the filter in the transmitter. To implement the raised-cosine response, for example, the filtering is split into two parts to create a matched set. When the raised-cosine filtering is split into two parts, each part is called the root-raised-cosine.